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Continuous Growth Models

Author(s): Anton E Weisstein

Truman State University

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Summary:
This Excel worksheet compares user-input growth data with predictions under linear, exponential, and logistic models of growth. Introduces concepts of statistical analysis that can be additionally explored with statistics software (e.g. JMP, SAS).

Licensed under CC Attribution-NonCommercial-ShareAlike 4.0 International according to these terms

Version 1.0 - published on 16 Aug 2024 doi:10.25334/PE67-JN93 - cite this

Keywords

  1. exponential growth
  2. logistic growth
  3. curve fitting
  4. linear growth
  5. growth models

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  3. Audience Level
  4. Software Used
Contents:
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Overview

This worksheet compares user-input growth data with predictions under linear, exponential, and logistic models of growth. Students can input parameters for each model; the program graphs the results and computes a crude goodness-of-fit measure. Introduces concepts of modeling and statistical analysis that can be more thoroughly explored using standard statistics software (JMP, SAS, etc.)

Popular Text Citations

Baker, G. L.; Gollub, J. P. 1990. "The logistic map" in: Chaotic Dynamics: an introduction. NY: Cambridge University Press.

Research Articles

Zwanzig, R. 1973. Generalized Verhulst Laws for Population Growth. Proc Natl Acad Sci U S A. 70 (11) 3048-3051.

Urszula Foryś, A. M. 2003. Logistic equations in tumor growth modeling. Int. J. Appl. Math. Comput. Sci 13:317-325.

Betty Tang and Gail S.K. Wolkowicz, (1992) "Mathematical Models of Microbial Growth and Competition in the Chemostat Regulated by Cell-Bound Extracellular Enzymes," Journal of Mathematical Biology 31, 1-23.

Julien Arino, Lin Wang, and Gail S. K. Wolkowicz, "An alternative formulation for a delayed logistic equation," to appear in Journal of Theoretical Biology.

Data Sources

United Nations. World Population Prospects: The 2004 Revision Population Database (Snapshot: 2007-01-13)

Education Research & Pedagogical Materials

Directions for simulating Verhulst equations on a TI-83 calculator

Steve Baedke, X-next logistic model, James Madison University, Department of Geology and Environmental Sciences (Snapshot: 2010-02-02)

Tutorial & Background Materials

T.J. Nelson, Population Dynamics (Snapshot: 2010-06-12)

Nardin Patrizia , The Verhulst Equation (Snapshot: 2013-04-13)

Citation

Researchers should cite this work as follows:

Fundamental Mathematical Concepts

Fundamental Mathematical Concepts
Logistic Function

Developed By

Developed by
Pierre Verhulst

Belgian mathematician who introduced the Verhulst equation (also known as the logistic equation ) to model human population growth in 1838. He quit his literary studies to devote himself to mathematics. As an undergraduate at the University of Ghent, he was awarded two academic prizes for his works on the calculus of variations. Later, he published papers on number theory and physics.

Primary Reference

Verhulst, P. F., (1838). Notice sur la loi que la population pursuit dans son accroissement. Corresp. Math. Phys. 10:113-121.

Verhulst, P. F. 1845. Recherches Mathematiques sur La Loi D'Accroissement de la Population, Nouveaux Memoires de l'Academie Royale des Sciences et Belles-Lettres de Bruxelles, 18, Art. 1, 1-45. (Mathematical Researches into the Law of Population Growth Increase)